1 (**************************************************************************)
4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.cs.unibo.it *)
10 (* \ / This file is distributed under the terms of the *)
11 (* v GNU General Public License Version 2 *)
13 (**************************************************************************)
15 (* This file was automatically generated: do not edit *********************)
17 include "LambdaDelta-1/clen/defs.ma".
19 include "LambdaDelta-1/getl/props.ma".
21 theorem getl_ctail_clen:
22 \forall (b: B).(\forall (t: T).(\forall (c: C).(ex nat (\lambda (n:
23 nat).(getl (clen c) (CTail (Bind b) t c) (CHead (CSort n) (Bind b) t))))))
25 \lambda (b: B).(\lambda (t: T).(\lambda (c: C).(C_ind (\lambda (c0: C).(ex
26 nat (\lambda (n: nat).(getl (clen c0) (CTail (Bind b) t c0) (CHead (CSort n)
27 (Bind b) t))))) (\lambda (n: nat).(ex_intro nat (\lambda (n0: nat).(getl O
28 (CHead (CSort n) (Bind b) t) (CHead (CSort n0) (Bind b) t))) n (getl_refl b
29 (CSort n) t))) (\lambda (c0: C).(\lambda (H: (ex nat (\lambda (n: nat).(getl
30 (clen c0) (CTail (Bind b) t c0) (CHead (CSort n) (Bind b) t))))).(\lambda (k:
31 K).(\lambda (t0: T).(let H0 \def H in (ex_ind nat (\lambda (n: nat).(getl
32 (clen c0) (CTail (Bind b) t c0) (CHead (CSort n) (Bind b) t))) (ex nat
33 (\lambda (n: nat).(getl (s k (clen c0)) (CHead (CTail (Bind b) t c0) k t0)
34 (CHead (CSort n) (Bind b) t)))) (\lambda (x: nat).(\lambda (H1: (getl (clen
35 c0) (CTail (Bind b) t c0) (CHead (CSort x) (Bind b) t))).(K_ind (\lambda (k0:
36 K).(ex nat (\lambda (n: nat).(getl (s k0 (clen c0)) (CHead (CTail (Bind b) t
37 c0) k0 t0) (CHead (CSort n) (Bind b) t))))) (\lambda (b0: B).(ex_intro nat
38 (\lambda (n: nat).(getl (S (clen c0)) (CHead (CTail (Bind b) t c0) (Bind b0)
39 t0) (CHead (CSort n) (Bind b) t))) x (getl_head (Bind b0) (clen c0) (CTail
40 (Bind b) t c0) (CHead (CSort x) (Bind b) t) H1 t0))) (\lambda (f:
41 F).(ex_intro nat (\lambda (n: nat).(getl (clen c0) (CHead (CTail (Bind b) t
42 c0) (Flat f) t0) (CHead (CSort n) (Bind b) t))) x (getl_flat (CTail (Bind b)
43 t c0) (CHead (CSort x) (Bind b) t) (clen c0) H1 f t0))) k))) H0)))))) c))).
45 theorem getl_gen_tail:
46 \forall (k: K).(\forall (b: B).(\forall (u1: T).(\forall (u2: T).(\forall
47 (c2: C).(\forall (c1: C).(\forall (i: nat).((getl i (CTail k u1 c1) (CHead c2
48 (Bind b) u2)) \to (or (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e)))
49 (\lambda (e: C).(getl i c1 (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_:
50 nat).(eq nat i (clen c1))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_:
51 nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c2 (CSort n))))))))))))
53 \lambda (k: K).(\lambda (b: B).(\lambda (u1: T).(\lambda (u2: T).(\lambda
54 (c2: C).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (i: nat).((getl i
55 (CTail k u1 c) (CHead c2 (Bind b) u2)) \to (or (ex2 C (\lambda (e: C).(eq C
56 c2 (CTail k u1 e))) (\lambda (e: C).(getl i c (CHead e (Bind b) u2)))) (ex4
57 nat (\lambda (_: nat).(eq nat i (clen c))) (\lambda (_: nat).(eq K k (Bind
58 b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c2 (CSort
59 n)))))))) (\lambda (n: nat).(\lambda (i: nat).(nat_ind (\lambda (n0:
60 nat).((getl n0 (CTail k u1 (CSort n)) (CHead c2 (Bind b) u2)) \to (or (ex2 C
61 (\lambda (e: C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl n0 (CSort n)
62 (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat n0 (clen (CSort
63 n)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2))
64 (\lambda (n1: nat).(eq C c2 (CSort n1))))))) (\lambda (H: (getl O (CHead
65 (CSort n) k u1) (CHead c2 (Bind b) u2))).(K_ind (\lambda (k0: K).((clear
66 (CHead (CSort n) k0 u1) (CHead c2 (Bind b) u2)) \to (or (ex2 C (\lambda (e:
67 C).(eq C c2 (CTail k0 u1 e))) (\lambda (e: C).(getl O (CSort n) (CHead e
68 (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat O O)) (\lambda (_:
69 nat).(eq K k0 (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n0:
70 nat).(eq C c2 (CSort n0))))))) (\lambda (b0: B).(\lambda (H0: (clear (CHead
71 (CSort n) (Bind b0) u1) (CHead c2 (Bind b) u2))).(let H1 \def (f_equal C C
72 (\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _)
73 \Rightarrow c2 | (CHead c _ _) \Rightarrow c])) (CHead c2 (Bind b) u2) (CHead
74 (CSort n) (Bind b0) u1) (clear_gen_bind b0 (CSort n) (CHead c2 (Bind b) u2)
75 u1 H0)) in ((let H2 \def (f_equal C B (\lambda (e: C).(match e in C return
76 (\lambda (_: C).B) with [(CSort _) \Rightarrow b | (CHead _ k0 _) \Rightarrow
77 (match k0 in K return (\lambda (_: K).B) with [(Bind b1) \Rightarrow b1 |
78 (Flat _) \Rightarrow b])])) (CHead c2 (Bind b) u2) (CHead (CSort n) (Bind b0)
79 u1) (clear_gen_bind b0 (CSort n) (CHead c2 (Bind b) u2) u1 H0)) in ((let H3
80 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T)
81 with [(CSort _) \Rightarrow u2 | (CHead _ _ t) \Rightarrow t])) (CHead c2
82 (Bind b) u2) (CHead (CSort n) (Bind b0) u1) (clear_gen_bind b0 (CSort n)
83 (CHead c2 (Bind b) u2) u1 H0)) in (\lambda (H4: (eq B b b0)).(\lambda (H5:
84 (eq C c2 (CSort n))).(eq_ind_r C (CSort n) (\lambda (c: C).(or (ex2 C
85 (\lambda (e: C).(eq C c (CTail (Bind b0) u1 e))) (\lambda (e: C).(getl O
86 (CSort n) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat O O))
87 (\lambda (_: nat).(eq K (Bind b0) (Bind b))) (\lambda (_: nat).(eq T u1 u2))
88 (\lambda (n0: nat).(eq C c (CSort n0)))))) (eq_ind_r T u1 (\lambda (t: T).(or
89 (ex2 C (\lambda (e: C).(eq C (CSort n) (CTail (Bind b0) u1 e))) (\lambda (e:
90 C).(getl O (CSort n) (CHead e (Bind b) t)))) (ex4 nat (\lambda (_: nat).(eq
91 nat O O)) (\lambda (_: nat).(eq K (Bind b0) (Bind b))) (\lambda (_: nat).(eq
92 T u1 t)) (\lambda (n0: nat).(eq C (CSort n) (CSort n0)))))) (eq_ind_r B b0
93 (\lambda (b1: B).(or (ex2 C (\lambda (e: C).(eq C (CSort n) (CTail (Bind b0)
94 u1 e))) (\lambda (e: C).(getl O (CSort n) (CHead e (Bind b1) u1)))) (ex4 nat
95 (\lambda (_: nat).(eq nat O O)) (\lambda (_: nat).(eq K (Bind b0) (Bind b1)))
96 (\lambda (_: nat).(eq T u1 u1)) (\lambda (n0: nat).(eq C (CSort n) (CSort
97 n0)))))) (or_intror (ex2 C (\lambda (e: C).(eq C (CSort n) (CTail (Bind b0)
98 u1 e))) (\lambda (e: C).(getl O (CSort n) (CHead e (Bind b0) u1)))) (ex4 nat
99 (\lambda (_: nat).(eq nat O O)) (\lambda (_: nat).(eq K (Bind b0) (Bind b0)))
100 (\lambda (_: nat).(eq T u1 u1)) (\lambda (n0: nat).(eq C (CSort n) (CSort
101 n0)))) (ex4_intro nat (\lambda (_: nat).(eq nat O O)) (\lambda (_: nat).(eq K
102 (Bind b0) (Bind b0))) (\lambda (_: nat).(eq T u1 u1)) (\lambda (n0: nat).(eq
103 C (CSort n) (CSort n0))) n (refl_equal nat O) (refl_equal K (Bind b0))
104 (refl_equal T u1) (refl_equal C (CSort n)))) b H4) u2 H3) c2 H5)))) H2))
105 H1)))) (\lambda (f: F).(\lambda (H0: (clear (CHead (CSort n) (Flat f) u1)
106 (CHead c2 (Bind b) u2))).(clear_gen_sort (CHead c2 (Bind b) u2) n
107 (clear_gen_flat f (CSort n) (CHead c2 (Bind b) u2) u1 H0) (or (ex2 C (\lambda
108 (e: C).(eq C c2 (CTail (Flat f) u1 e))) (\lambda (e: C).(getl O (CSort n)
109 (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat O O)) (\lambda
110 (_: nat).(eq K (Flat f) (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda
111 (n0: nat).(eq C c2 (CSort n0)))))))) k (getl_gen_O (CHead (CSort n) k u1)
112 (CHead c2 (Bind b) u2) H))) (\lambda (n0: nat).(\lambda (_: (((getl n0 (CHead
113 (CSort n) k u1) (CHead c2 (Bind b) u2)) \to (or (ex2 C (\lambda (e: C).(eq C
114 c2 (CTail k u1 e))) (\lambda (e: C).(getl n0 (CSort n) (CHead e (Bind b)
115 u2)))) (ex4 nat (\lambda (_: nat).(eq nat n0 O)) (\lambda (_: nat).(eq K k
116 (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n1: nat).(eq C c2 (CSort
117 n1)))))))).(\lambda (H0: (getl (S n0) (CHead (CSort n) k u1) (CHead c2 (Bind
118 b) u2))).(getl_gen_sort n (r k n0) (CHead c2 (Bind b) u2) (getl_gen_S k
119 (CSort n) (CHead c2 (Bind b) u2) u1 n0 H0) (or (ex2 C (\lambda (e: C).(eq C
120 c2 (CTail k u1 e))) (\lambda (e: C).(getl (S n0) (CSort n) (CHead e (Bind b)
121 u2)))) (ex4 nat (\lambda (_: nat).(eq nat (S n0) O)) (\lambda (_: nat).(eq K
122 k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n1: nat).(eq C c2
123 (CSort n1))))))))) i))) (\lambda (c: C).(\lambda (H: ((\forall (i:
124 nat).((getl i (CTail k u1 c) (CHead c2 (Bind b) u2)) \to (or (ex2 C (\lambda
125 (e: C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl i c (CHead e (Bind b)
126 u2)))) (ex4 nat (\lambda (_: nat).(eq nat i (clen c))) (\lambda (_: nat).(eq
127 K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c2
128 (CSort n))))))))).(\lambda (k0: K).(\lambda (t: T).(\lambda (i: nat).(nat_ind
129 (\lambda (n: nat).((getl n (CTail k u1 (CHead c k0 t)) (CHead c2 (Bind b)
130 u2)) \to (or (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e))) (\lambda (e:
131 C).(getl n (CHead c k0 t) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_:
132 nat).(eq nat n (clen (CHead c k0 t)))) (\lambda (_: nat).(eq K k (Bind b)))
133 (\lambda (_: nat).(eq T u1 u2)) (\lambda (n0: nat).(eq C c2 (CSort n0)))))))
134 (\lambda (H0: (getl O (CHead (CTail k u1 c) k0 t) (CHead c2 (Bind b)
135 u2))).(K_ind (\lambda (k1: K).((clear (CHead (CTail k u1 c) k1 t) (CHead c2
136 (Bind b) u2)) \to (or (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e)))
137 (\lambda (e: C).(getl O (CHead c k1 t) (CHead e (Bind b) u2)))) (ex4 nat
138 (\lambda (_: nat).(eq nat O (s k1 (clen c)))) (\lambda (_: nat).(eq K k (Bind
139 b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c2 (CSort
140 n))))))) (\lambda (b0: B).(\lambda (H1: (clear (CHead (CTail k u1 c) (Bind
141 b0) t) (CHead c2 (Bind b) u2))).(let H2 \def (f_equal C C (\lambda (e:
142 C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c2 |
143 (CHead c0 _ _) \Rightarrow c0])) (CHead c2 (Bind b) u2) (CHead (CTail k u1 c)
144 (Bind b0) t) (clear_gen_bind b0 (CTail k u1 c) (CHead c2 (Bind b) u2) t H1))
145 in ((let H3 \def (f_equal C B (\lambda (e: C).(match e in C return (\lambda
146 (_: C).B) with [(CSort _) \Rightarrow b | (CHead _ k1 _) \Rightarrow (match
147 k1 in K return (\lambda (_: K).B) with [(Bind b1) \Rightarrow b1 | (Flat _)
148 \Rightarrow b])])) (CHead c2 (Bind b) u2) (CHead (CTail k u1 c) (Bind b0) t)
149 (clear_gen_bind b0 (CTail k u1 c) (CHead c2 (Bind b) u2) t H1)) in ((let H4
150 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T)
151 with [(CSort _) \Rightarrow u2 | (CHead _ _ t0) \Rightarrow t0])) (CHead c2
152 (Bind b) u2) (CHead (CTail k u1 c) (Bind b0) t) (clear_gen_bind b0 (CTail k
153 u1 c) (CHead c2 (Bind b) u2) t H1)) in (\lambda (H5: (eq B b b0)).(\lambda
154 (H6: (eq C c2 (CTail k u1 c))).(eq_ind T u2 (\lambda (t0: T).(or (ex2 C
155 (\lambda (e: C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl O (CHead c
156 (Bind b0) t0) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat O
157 (s (Bind b0) (clen c)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_:
158 nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c2 (CSort n)))))) (eq_ind B b
159 (\lambda (b1: B).(or (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e)))
160 (\lambda (e: C).(getl O (CHead c (Bind b1) u2) (CHead e (Bind b) u2)))) (ex4
161 nat (\lambda (_: nat).(eq nat O (s (Bind b1) (clen c)))) (\lambda (_:
162 nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq
163 C c2 (CSort n)))))) (let H7 \def (eq_ind C c2 (\lambda (c0: C).(\forall (i0:
164 nat).((getl i0 (CTail k u1 c) (CHead c0 (Bind b) u2)) \to (or (ex2 C (\lambda
165 (e: C).(eq C c0 (CTail k u1 e))) (\lambda (e: C).(getl i0 c (CHead e (Bind b)
166 u2)))) (ex4 nat (\lambda (_: nat).(eq nat i0 (clen c))) (\lambda (_: nat).(eq
167 K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c0
168 (CSort n)))))))) H (CTail k u1 c) H6) in (eq_ind_r C (CTail k u1 c) (\lambda
169 (c0: C).(or (ex2 C (\lambda (e: C).(eq C c0 (CTail k u1 e))) (\lambda (e:
170 C).(getl O (CHead c (Bind b) u2) (CHead e (Bind b) u2)))) (ex4 nat (\lambda
171 (_: nat).(eq nat O (s (Bind b) (clen c)))) (\lambda (_: nat).(eq K k (Bind
172 b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c0 (CSort
173 n)))))) (or_introl (ex2 C (\lambda (e: C).(eq C (CTail k u1 c) (CTail k u1
174 e))) (\lambda (e: C).(getl O (CHead c (Bind b) u2) (CHead e (Bind b) u2))))
175 (ex4 nat (\lambda (_: nat).(eq nat O (s (Bind b) (clen c)))) (\lambda (_:
176 nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq
177 C (CTail k u1 c) (CSort n)))) (ex_intro2 C (\lambda (e: C).(eq C (CTail k u1
178 c) (CTail k u1 e))) (\lambda (e: C).(getl O (CHead c (Bind b) u2) (CHead e
179 (Bind b) u2))) c (refl_equal C (CTail k u1 c)) (getl_refl b c u2))) c2 H6))
180 b0 H5) t H4)))) H3)) H2)))) (\lambda (f: F).(\lambda (H1: (clear (CHead
181 (CTail k u1 c) (Flat f) t) (CHead c2 (Bind b) u2))).(let H2 \def (H O
182 (getl_intro O (CTail k u1 c) (CHead c2 (Bind b) u2) (CTail k u1 c) (drop_refl
183 (CTail k u1 c)) (clear_gen_flat f (CTail k u1 c) (CHead c2 (Bind b) u2) t
184 H1))) in (or_ind (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e))) (\lambda
185 (e: C).(getl O c (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat
186 O (clen c))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1
187 u2)) (\lambda (n: nat).(eq C c2 (CSort n)))) (or (ex2 C (\lambda (e: C).(eq C
188 c2 (CTail k u1 e))) (\lambda (e: C).(getl O (CHead c (Flat f) t) (CHead e
189 (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat O (s (Flat f) (clen c))))
190 (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda
191 (n: nat).(eq C c2 (CSort n))))) (\lambda (H3: (ex2 C (\lambda (e: C).(eq C c2
192 (CTail k u1 e))) (\lambda (e: C).(getl O c (CHead e (Bind b) u2))))).(ex2_ind
193 C (\lambda (e: C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl O c (CHead
194 e (Bind b) u2))) (or (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e)))
195 (\lambda (e: C).(getl O (CHead c (Flat f) t) (CHead e (Bind b) u2)))) (ex4
196 nat (\lambda (_: nat).(eq nat O (s (Flat f) (clen c)))) (\lambda (_: nat).(eq
197 K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c2
198 (CSort n))))) (\lambda (x: C).(\lambda (H4: (eq C c2 (CTail k u1
199 x))).(\lambda (H5: (getl O c (CHead x (Bind b) u2))).(eq_ind_r C (CTail k u1
200 x) (\lambda (c0: C).(or (ex2 C (\lambda (e: C).(eq C c0 (CTail k u1 e)))
201 (\lambda (e: C).(getl O (CHead c (Flat f) t) (CHead e (Bind b) u2)))) (ex4
202 nat (\lambda (_: nat).(eq nat O (s (Flat f) (clen c)))) (\lambda (_: nat).(eq
203 K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c0
204 (CSort n)))))) (or_introl (ex2 C (\lambda (e: C).(eq C (CTail k u1 x) (CTail
205 k u1 e))) (\lambda (e: C).(getl O (CHead c (Flat f) t) (CHead e (Bind b)
206 u2)))) (ex4 nat (\lambda (_: nat).(eq nat O (s (Flat f) (clen c)))) (\lambda
207 (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n:
208 nat).(eq C (CTail k u1 x) (CSort n)))) (ex_intro2 C (\lambda (e: C).(eq C
209 (CTail k u1 x) (CTail k u1 e))) (\lambda (e: C).(getl O (CHead c (Flat f) t)
210 (CHead e (Bind b) u2))) x (refl_equal C (CTail k u1 x)) (getl_flat c (CHead x
211 (Bind b) u2) O H5 f t))) c2 H4)))) H3)) (\lambda (H3: (ex4 nat (\lambda (_:
212 nat).(eq nat O (clen c))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_:
213 nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c2 (CSort n))))).(ex4_ind nat
214 (\lambda (_: nat).(eq nat O (clen c))) (\lambda (_: nat).(eq K k (Bind b)))
215 (\lambda (_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c2 (CSort n))) (or
216 (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl O
217 (CHead c (Flat f) t) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq
218 nat O (s (Flat f) (clen c)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda
219 (_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c2 (CSort n))))) (\lambda (x0:
220 nat).(\lambda (H4: (eq nat O (clen c))).(\lambda (H5: (eq K k (Bind
221 b))).(\lambda (H6: (eq T u1 u2)).(\lambda (H7: (eq C c2 (CSort
222 x0))).(eq_ind_r C (CSort x0) (\lambda (c0: C).(or (ex2 C (\lambda (e: C).(eq
223 C c0 (CTail k u1 e))) (\lambda (e: C).(getl O (CHead c (Flat f) t) (CHead e
224 (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat O (s (Flat f) (clen c))))
225 (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda
226 (n: nat).(eq C c0 (CSort n)))))) (eq_ind T u1 (\lambda (t0: T).(or (ex2 C
227 (\lambda (e: C).(eq C (CSort x0) (CTail k u1 e))) (\lambda (e: C).(getl O
228 (CHead c (Flat f) t) (CHead e (Bind b) t0)))) (ex4 nat (\lambda (_: nat).(eq
229 nat O (s (Flat f) (clen c)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda
230 (_: nat).(eq T u1 t0)) (\lambda (n: nat).(eq C (CSort x0) (CSort n))))))
231 (eq_ind_r K (Bind b) (\lambda (k1: K).(or (ex2 C (\lambda (e: C).(eq C (CSort
232 x0) (CTail k1 u1 e))) (\lambda (e: C).(getl O (CHead c (Flat f) t) (CHead e
233 (Bind b) u1)))) (ex4 nat (\lambda (_: nat).(eq nat O (s (Flat f) (clen c))))
234 (\lambda (_: nat).(eq K k1 (Bind b))) (\lambda (_: nat).(eq T u1 u1))
235 (\lambda (n: nat).(eq C (CSort x0) (CSort n)))))) (or_intror (ex2 C (\lambda
236 (e: C).(eq C (CSort x0) (CTail (Bind b) u1 e))) (\lambda (e: C).(getl O
237 (CHead c (Flat f) t) (CHead e (Bind b) u1)))) (ex4 nat (\lambda (_: nat).(eq
238 nat O (s (Flat f) (clen c)))) (\lambda (_: nat).(eq K (Bind b) (Bind b)))
239 (\lambda (_: nat).(eq T u1 u1)) (\lambda (n: nat).(eq C (CSort x0) (CSort
240 n)))) (ex4_intro nat (\lambda (_: nat).(eq nat O (s (Flat f) (clen c))))
241 (\lambda (_: nat).(eq K (Bind b) (Bind b))) (\lambda (_: nat).(eq T u1 u1))
242 (\lambda (n: nat).(eq C (CSort x0) (CSort n))) x0 H4 (refl_equal K (Bind b))
243 (refl_equal T u1) (refl_equal C (CSort x0)))) k H5) u2 H6) c2 H7)))))) H3))
244 H2)))) k0 (getl_gen_O (CHead (CTail k u1 c) k0 t) (CHead c2 (Bind b) u2)
245 H0))) (\lambda (n: nat).(\lambda (H0: (((getl n (CHead (CTail k u1 c) k0 t)
246 (CHead c2 (Bind b) u2)) \to (or (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1
247 e))) (\lambda (e: C).(getl n (CHead c k0 t) (CHead e (Bind b) u2)))) (ex4 nat
248 (\lambda (_: nat).(eq nat n (s k0 (clen c)))) (\lambda (_: nat).(eq K k (Bind
249 b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n0: nat).(eq C c2 (CSort
250 n0)))))))).(\lambda (H1: (getl (S n) (CHead (CTail k u1 c) k0 t) (CHead c2
251 (Bind b) u2))).(let H_x \def (H (r k0 n) (getl_gen_S k0 (CTail k u1 c) (CHead
252 c2 (Bind b) u2) t n H1)) in (let H2 \def H_x in (or_ind (ex2 C (\lambda (e:
253 C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl (r k0 n) c (CHead e (Bind
254 b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat (r k0 n) (clen c))) (\lambda (_:
255 nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n0:
256 nat).(eq C c2 (CSort n0)))) (or (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1
257 e))) (\lambda (e: C).(getl (S n) (CHead c k0 t) (CHead e (Bind b) u2)))) (ex4
258 nat (\lambda (_: nat).(eq nat (S n) (s k0 (clen c)))) (\lambda (_: nat).(eq K
259 k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n0: nat).(eq C c2
260 (CSort n0))))) (\lambda (H3: (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e)))
261 (\lambda (e: C).(getl (r k0 n) c (CHead e (Bind b) u2))))).(ex2_ind C
262 (\lambda (e: C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl (r k0 n) c
263 (CHead e (Bind b) u2))) (or (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e)))
264 (\lambda (e: C).(getl (S n) (CHead c k0 t) (CHead e (Bind b) u2)))) (ex4 nat
265 (\lambda (_: nat).(eq nat (S n) (s k0 (clen c)))) (\lambda (_: nat).(eq K k
266 (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n0: nat).(eq C c2 (CSort
267 n0))))) (\lambda (x: C).(\lambda (H4: (eq C c2 (CTail k u1 x))).(\lambda (H5:
268 (getl (r k0 n) c (CHead x (Bind b) u2))).(let H6 \def (eq_ind C c2 (\lambda
269 (c0: C).(getl (r k0 n) (CTail k u1 c) (CHead c0 (Bind b) u2))) (getl_gen_S k0
270 (CTail k u1 c) (CHead c2 (Bind b) u2) t n H1) (CTail k u1 x) H4) in (let H7
271 \def (eq_ind C c2 (\lambda (c0: C).((getl n (CHead (CTail k u1 c) k0 t)
272 (CHead c0 (Bind b) u2)) \to (or (ex2 C (\lambda (e: C).(eq C c0 (CTail k u1
273 e))) (\lambda (e: C).(getl n (CHead c k0 t) (CHead e (Bind b) u2)))) (ex4 nat
274 (\lambda (_: nat).(eq nat n (s k0 (clen c)))) (\lambda (_: nat).(eq K k (Bind
275 b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n0: nat).(eq C c0 (CSort
276 n0))))))) H0 (CTail k u1 x) H4) in (eq_ind_r C (CTail k u1 x) (\lambda (c0:
277 C).(or (ex2 C (\lambda (e: C).(eq C c0 (CTail k u1 e))) (\lambda (e: C).(getl
278 (S n) (CHead c k0 t) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq
279 nat (S n) (s k0 (clen c)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_:
280 nat).(eq T u1 u2)) (\lambda (n0: nat).(eq C c0 (CSort n0)))))) (or_introl
281 (ex2 C (\lambda (e: C).(eq C (CTail k u1 x) (CTail k u1 e))) (\lambda (e:
282 C).(getl (S n) (CHead c k0 t) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_:
283 nat).(eq nat (S n) (s k0 (clen c)))) (\lambda (_: nat).(eq K k (Bind b)))
284 (\lambda (_: nat).(eq T u1 u2)) (\lambda (n0: nat).(eq C (CTail k u1 x)
285 (CSort n0)))) (ex_intro2 C (\lambda (e: C).(eq C (CTail k u1 x) (CTail k u1
286 e))) (\lambda (e: C).(getl (S n) (CHead c k0 t) (CHead e (Bind b) u2))) x
287 (refl_equal C (CTail k u1 x)) (getl_head k0 n c (CHead x (Bind b) u2) H5 t)))
288 c2 H4)))))) H3)) (\lambda (H3: (ex4 nat (\lambda (_: nat).(eq nat (r k0 n)
289 (clen c))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1
290 u2)) (\lambda (n0: nat).(eq C c2 (CSort n0))))).(ex4_ind nat (\lambda (_:
291 nat).(eq nat (r k0 n) (clen c))) (\lambda (_: nat).(eq K k (Bind b)))
292 (\lambda (_: nat).(eq T u1 u2)) (\lambda (n0: nat).(eq C c2 (CSort n0))) (or
293 (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl (S n)
294 (CHead c k0 t) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat (S
295 n) (s k0 (clen c)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_:
296 nat).(eq T u1 u2)) (\lambda (n0: nat).(eq C c2 (CSort n0))))) (\lambda (x0:
297 nat).(\lambda (H4: (eq nat (r k0 n) (clen c))).(\lambda (H5: (eq K k (Bind
298 b))).(\lambda (H6: (eq T u1 u2)).(\lambda (H7: (eq C c2 (CSort x0))).(let H8
299 \def (eq_ind C c2 (\lambda (c0: C).(getl (r k0 n) (CTail k u1 c) (CHead c0
300 (Bind b) u2))) (getl_gen_S k0 (CTail k u1 c) (CHead c2 (Bind b) u2) t n H1)
301 (CSort x0) H7) in (let H9 \def (eq_ind C c2 (\lambda (c0: C).((getl n (CHead
302 (CTail k u1 c) k0 t) (CHead c0 (Bind b) u2)) \to (or (ex2 C (\lambda (e:
303 C).(eq C c0 (CTail k u1 e))) (\lambda (e: C).(getl n (CHead c k0 t) (CHead e
304 (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat n (s k0 (clen c))))
305 (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda
306 (n0: nat).(eq C c0 (CSort n0))))))) H0 (CSort x0) H7) in (eq_ind_r C (CSort
307 x0) (\lambda (c0: C).(or (ex2 C (\lambda (e: C).(eq C c0 (CTail k u1 e)))
308 (\lambda (e: C).(getl (S n) (CHead c k0 t) (CHead e (Bind b) u2)))) (ex4 nat
309 (\lambda (_: nat).(eq nat (S n) (s k0 (clen c)))) (\lambda (_: nat).(eq K k
310 (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n0: nat).(eq C c0 (CSort
311 n0)))))) (let H10 \def (eq_ind_r T u2 (\lambda (t0: T).((getl n (CHead (CTail
312 k u1 c) k0 t) (CHead (CSort x0) (Bind b) t0)) \to (or (ex2 C (\lambda (e:
313 C).(eq C (CSort x0) (CTail k u1 e))) (\lambda (e: C).(getl n (CHead c k0 t)
314 (CHead e (Bind b) t0)))) (ex4 nat (\lambda (_: nat).(eq nat n (s k0 (clen
315 c)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 t0))
316 (\lambda (n0: nat).(eq C (CSort x0) (CSort n0))))))) H9 u1 H6) in (let H11
317 \def (eq_ind_r T u2 (\lambda (t0: T).(getl (r k0 n) (CTail k u1 c) (CHead
318 (CSort x0) (Bind b) t0))) H8 u1 H6) in (eq_ind T u1 (\lambda (t0: T).(or (ex2
319 C (\lambda (e: C).(eq C (CSort x0) (CTail k u1 e))) (\lambda (e: C).(getl (S
320 n) (CHead c k0 t) (CHead e (Bind b) t0)))) (ex4 nat (\lambda (_: nat).(eq nat
321 (S n) (s k0 (clen c)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_:
322 nat).(eq T u1 t0)) (\lambda (n0: nat).(eq C (CSort x0) (CSort n0)))))) (let
323 H12 \def (eq_ind K k (\lambda (k1: K).((getl n (CHead (CTail k1 u1 c) k0 t)
324 (CHead (CSort x0) (Bind b) u1)) \to (or (ex2 C (\lambda (e: C).(eq C (CSort
325 x0) (CTail k1 u1 e))) (\lambda (e: C).(getl n (CHead c k0 t) (CHead e (Bind
326 b) u1)))) (ex4 nat (\lambda (_: nat).(eq nat n (s k0 (clen c)))) (\lambda (_:
327 nat).(eq K k1 (Bind b))) (\lambda (_: nat).(eq T u1 u1)) (\lambda (n0:
328 nat).(eq C (CSort x0) (CSort n0))))))) H10 (Bind b) H5) in (let H13 \def
329 (eq_ind K k (\lambda (k1: K).(getl (r k0 n) (CTail k1 u1 c) (CHead (CSort x0)
330 (Bind b) u1))) H11 (Bind b) H5) in (eq_ind_r K (Bind b) (\lambda (k1: K).(or
331 (ex2 C (\lambda (e: C).(eq C (CSort x0) (CTail k1 u1 e))) (\lambda (e:
332 C).(getl (S n) (CHead c k0 t) (CHead e (Bind b) u1)))) (ex4 nat (\lambda (_:
333 nat).(eq nat (S n) (s k0 (clen c)))) (\lambda (_: nat).(eq K k1 (Bind b)))
334 (\lambda (_: nat).(eq T u1 u1)) (\lambda (n0: nat).(eq C (CSort x0) (CSort
335 n0)))))) (eq_ind nat (r k0 n) (\lambda (n0: nat).(or (ex2 C (\lambda (e:
336 C).(eq C (CSort x0) (CTail (Bind b) u1 e))) (\lambda (e: C).(getl (S n)
337 (CHead c k0 t) (CHead e (Bind b) u1)))) (ex4 nat (\lambda (_: nat).(eq nat (S
338 n) (s k0 n0))) (\lambda (_: nat).(eq K (Bind b) (Bind b))) (\lambda (_:
339 nat).(eq T u1 u1)) (\lambda (n1: nat).(eq C (CSort x0) (CSort n1))))))
340 (eq_ind_r nat (S n) (\lambda (n0: nat).(or (ex2 C (\lambda (e: C).(eq C
341 (CSort x0) (CTail (Bind b) u1 e))) (\lambda (e: C).(getl (S n) (CHead c k0 t)
342 (CHead e (Bind b) u1)))) (ex4 nat (\lambda (_: nat).(eq nat (S n) n0))
343 (\lambda (_: nat).(eq K (Bind b) (Bind b))) (\lambda (_: nat).(eq T u1 u1))
344 (\lambda (n1: nat).(eq C (CSort x0) (CSort n1)))))) (or_intror (ex2 C
345 (\lambda (e: C).(eq C (CSort x0) (CTail (Bind b) u1 e))) (\lambda (e:
346 C).(getl (S n) (CHead c k0 t) (CHead e (Bind b) u1)))) (ex4 nat (\lambda (_:
347 nat).(eq nat (S n) (S n))) (\lambda (_: nat).(eq K (Bind b) (Bind b)))
348 (\lambda (_: nat).(eq T u1 u1)) (\lambda (n0: nat).(eq C (CSort x0) (CSort
349 n0)))) (ex4_intro nat (\lambda (_: nat).(eq nat (S n) (S n))) (\lambda (_:
350 nat).(eq K (Bind b) (Bind b))) (\lambda (_: nat).(eq T u1 u1)) (\lambda (n0:
351 nat).(eq C (CSort x0) (CSort n0))) x0 (refl_equal nat (S n)) (refl_equal K
352 (Bind b)) (refl_equal T u1) (refl_equal C (CSort x0)))) (s k0 (r k0 n)) (s_r
353 k0 n)) (clen c) H4) k H5))) u2 H6))) c2 H7)))))))) H3)) H2)))))) i))))))